Context. A method for constructing an algorithm for stabilizing the interoperability of a production line is considered. The object of the study was a model of a multi-operational production line.

Objective. The goal of the work is to develop a method for constructing an optimal algorithm for stabilizing the flow parameters of a production line, which provides asymptotic stability of the state of flow parameters for a given quality of the process.

Method. A method for constructing an algorithm for stabilizing the level of interoperative backlogs of a multi-operational production line is proposed. The stabilization algorithm is based on a two-moment PDE-model of the production line, which made it possible to represent the production line in the form of a complex dynamic distributed system. This representation made it possible to define the stabilizing control in the form of a function that depends not only on time but also on the coordinates characterizing the location of technological equipment along the production line. The use of the method of Lyapunov functions made it possible to synthesize the optimal stabilizing control of the state of interoperation backlogs at technological operations of the production line, which ensures the asymptotic stability of the  given unperturbed state of the flow parameters of the production line at the lowest cost of technological resources spent on the formation of the control action. The requirement for the best quality of the transition process from a disturbed state to an unperturbed state is expressed by the quality integral, which depends both on the magnitude of the disturbances that have arisen and on the magnitude of the stabilizing controls aimed at eliminating these disturbances.

Results. On the basis of the developed method for constructing an algorithm for stabilizing the state of flow parameters of a production line, an algorithm for stabilizing the value of interoperation backlogs at technological operations of a production line is synthesized.

Conclusions. The use of the method of Lyapunov functions in the synthesis of optimal stabilizing control of the flow parameters of the production line makes it possible to provide asymptotic damping of the arising disturbances of the flow parameters with the least cost of technological resources spent on the formation of the control action. It is shown that in the problem of stabilizing the state of interoperative backlogs, the stabilizing value of the control is proportional to the value of the arising disturbance. The proportionality coefficient is determined through the coefficients of the quality integral and the Lyapunov function. The prospect of further research is the development of a method for constructing an algorithm for stabilizing the productivity of technological operations of a production line.

Vollmann T. E., Berry L., Jacobs F. R. Manufacturing Planning and Control for Supply Chain Management. New York, McGraw-Hill, 2005, 520 p

Perdaen D., Armbruster D., Kempf K., Lefeber E. Controlling a re-entrant manufacturing line via the push-pull point, International Journal of Production Research, 2008, № 46 (16), pp. 4521–4536. DOI: 10.1080/00207540701258051

Forrester J. W. Industrial Dynamics. Cambridge, MIT. Press, 1961, 464 p

Pihnastyi, O. M. Statistical theory of control systems of the flow production. Lap Lambert Academic Publishing, 2018, 436 p

Armbruster D., Ringhofer C., Jo T. J. Continuous models for production flows, American Control Conference. Boston, MA, USA, 2004. №5, pp. 4589–4594. DOI: 10.1109/ACC.2004.182675

Schmitz J. P., Beek D. A., Rooda J. E. Chaos in Discrete Production Systems? Journal of Manufacturing Systems, 2002, No. 21(3), pp. 236–246. DOI: 10.1016/S02786125(02)80164-9

Bramson M. Stability of queueing networks, lecture notes in mathematics, Journal of Probability Surveys, 2008, No. 5. – pp. 169–345. DOI:10.1214/08-PS137. https://projecteuclid.org/euclid.ps/1220879338

Tian F., Willems S. P., Kempf K. G. An iterative approach to item-level tactical production and inventory planning, International Journal of Production Economics, 2011, No. 133, pp. 439–450 DOI:10.1016/j.ijpe.2010.07.011

Liang Z. System-theoretic properties of Production Lines, Doctoral dissertation, University of Michigan, 2009, 274 p. http://hdl.handle.net/2027.42/63812

Pihnastyi O. Statistical validity and derivation of balance equations for the two-level model of a production line, Eastern-European Journal of Enterprise Technologies, 2016, No. 5(4(83)), pp. 17–22. DOI: 10.15587/17294061.2016.81308

Kempf K., Uzsoy R., Smith St., Gary K. Evaluation and comparison of production schedules, Computers in Industry, 2000, No. 42(2–3), pp. 203–220. DOI: 10.1016/S01663615(99)00071-8

Cassandras C. G., Wardi Y., Melamed B., Sun Gang, Panayiotou C. G. Perturbation analysis for online control and optimization of stochastic fluid models, Transactions on Automatic Control, 2002. No. 47(8), pp. 1234–1248. DOI:10.1109/TAC.2002.800739.

Christofides P., Daoutidis P. Robust control of hyperbolic PDE systems, Chemical Engineering Science, 1998, No. 53(1), pp. 85–105.

Karmarkar U.S. Capacity Loading and Release Planning with Work-in-Progress (WIP) and Leadtimes, Journal of Manufacturing and Operations Management, 1989, No. 2, pp. 105– 123. https://www.iaorifors.com/paper/3710

Selçuk B., Fransoo J., Kok A. Work-in-process clearing in supply chain operations planning, IIE Transactions, 2006, No. 40(3), pp. 206–220. DOI: 10.1080/07408170701487997

Gross D., Harris C. M. Fundamentals of Queueing Theory. New York, 1974, 490 p.

Ramadge P. J., Wonham W. M. The control of discrete event systems, Proceedings of IEEE, 1989, № 77(1),. –P. 81-98. DOI: 10.1109/5.21072

Wonham W., Cai K., Rudie K. Supervisory Control of Discrete-Event Systems: A Brief History – 1980-2015, 20th IFAC World Congress IFAC-PapersOnLine, 2017, No. 50(1), pp. 1791–1797. DOI: 10.1016/j.ifacol.2017.08.164

Pihnastyi O., Khodusov V. D.Optimal Control Problem for a Conveyor-Type Production Line, Cybern. Syst. Anal., 2018, No. 54 (5), pp.744–753. DOI: 10.1007/s10559-018-0076-2

Pihnastyi O. M., Yemelianova D., Lysytsia D. Using pde model and system dynamics model for describing multioperation production lines, Eastern-European Journal of Enterprise Technologies, 2020, №4(4(106)), pp. 54–60. DOI: 10.15587/1729-4061.2020.210750

Takeda H. The Synchronized Production System. London, Kogan Page Ltd, 2006, 264 p.

Chankov S., Hütt M., Bendul J. Influencing factors of synchronization in manufacturing systems, International Journal of Production Research, 2018, No. 56(14), pp. 4781– 4801. DOI: 10.1080/00207543.2017.1400707

Kempf K. Simulating semiconductor manufacturing systems: successes, failures and deep questions / K.Kempf // Proceedings of the 1996 Winter Simulation Conference, Institute of Electrical and Electronics Engineers. Piscataway, New Jersey, 1996, pp. 3–11. DOI: 10.1109/WSC.1996.873254

Bitran G., Haas E., Hax A. Hierarchical production planning: a two-stage system, Operations Research, 1982, № 30, pp. 232–251. DOI: 10.1287/opre.30.2.232

Pihnastyi O. M., Khodusov V. D. Stochastic equation of the technological process, International Conference on System analysis & Intelligent computing, 2018, pp. 64–67. DOI:10.1109/SAIC.2018.8516833

1. Vollmann T. E. Manufacturing Planning and Control for Supply Chain Management. / T. E. Vollmann, L. Berry, F. R. Jacobs. – New York : McGraw-Hill, 2005. –520 p

2. Perdaen D. Controlling a re-entrant manufacturing line via the push-pull point. / D. Perdaen, D. Armbruster, K. Kempf, E. Lefeber // International Journal of Production Research. – 2008. – № 46 (16). – P.4521–4536. DOI: 10.1080/00207540701258051

3. Forrester J. W. Industrial Dynamics / J. W. Forrester. – Cambridge : MIT, Press, 1961. –464 p

4. Pihnastyi O. M. Statistical theory of control systems of the flow production / O. M. Pihnastyi. – Lap Lambert Academic Publishing, 2018. – 436 p

5. Armbruster D. Continuous models for production flows / D. Armbruster, C. Ringhofer, T. J. Jo // American Control Conference. Boston, MA, USA. – 2004. – № 5. – P. 4589– 4594. DOI: 10.1109/ACC.2004.182675

6. Schmitz J. P. Chaos in Discrete Production Systems? / J. P. Schmitz, D. A. Beek, J. E. Rooda // Journal of Manufacturing Systems. –2002. – №21 (3). – P. 236–246. DOI: 10.1016/S0278-6125(02)80164-9

7. Bramson M. Stability of queueing networks, lecture notes in mathematics / M. Bramson // Journal of Probability Surveys. – 2008. – № 5. – P. 169–345. DOI: 10.1214/08-PS137. https://projecteuclid.org/euclid.ps/1220879338

8. Tian F. An iterative approach to item-level tactical production and inventory planning / F. Tian, S. P. Willems, K. G. Kempf // International Journal of Production Economics, – 2011. – № 133. – P.439–450.  DOI: 10.1016/j.ijpe.2010.07.011

9. Liang Z. System-theoretic properties of Production Lines / Z. Liang // Doctoral dissertation, University of Michigan. 2009. –274 p.  http://hdl.handle.net/2027.42/63812

10. Pihnastyi O. Statistical validity and derivation of balance equations for the two-level model of a production line / O. Pihnastyi // Eastern-European Journal of Enterprise Technologies. – 2016. – 5(4(83)). – P. 17–22. DOI: 10.15587/1729-4061.2016.81308

11. Evaluation and comparison of production schedules / [K. Kempf, R. Uzsoy, St. Smith, K. Gary] // Computers in Industry. – 2000. – 42(2–3). – P. 203–220. DOI: 10.1016/S0166-3615(99)00071-8

12. Cassandras C. G. Perturbation analysis for online control and optimization of stochastic fluid models / [C. G. Cassandras, Y. Wardi, B. Melamed et al.]  // Transactions on Automatic Control. – 2002. – № 47 (8). – P. 1234–1248.  DOI: 10.1109/TAC.2002.800739.

13. Christofides P. Robust control of hyperbolic PDE systems / P. Christofides, P. Daoutidis // Chemical Engineering Science. –1998. – №53(1). –P. 85–105.

14. Karmarkar U. S. Capacity Loading and Release Planning with Work-in-Progress (WIP) and Leadtimes / U. S. Karmarkar // Journal of Manufacturing and Operations Management. – 1989. – № 2. – P. 105–123. https://www.iaorifors.com/paper/3710

15. Selçuk B. Work-in-process clearing in supply chain operations planning / B. Selçuk, J. Fransoo, A. Kok // IIE Transactions. – 2006. – № 40(3). – P. 206–220. DOI: 10.1080/07408170701487997   

16. Gross D. Fundamentals of Queueing Theory / D. Gross, C. M. Harris. – New York, 1974. –490 p.

17. Ramadge P. J. The control of discrete event systems / P. J. Ramadge, W. M. Wonham // Proceedings of IEEE. – 1989. – № 77(1). – P. 81–98. DOI: 10.1109/5.21072

18. Wonham W. Supervisory Control of Discrete-Event Systems: A Brief History – 1980-2015 / W. Wonham, K. Cai, K. Rudie// 20th IFAC World Congress IFAC-PapersOnLine. –2017. – № 50(1). – P. 1791–1797. DOI: 10.1016/j.ifacol.2017.08.164

19. Pihnastyi O.  Optimal Control Problem for a Conveyor-Type Production Line / O. M. Pihnastyi, V. D. Khodusov // Cybern. Syst. Anal. – 2018. – №54(5). – P.744–753.  DOI: 10.1007/s10559-018-0076-2

20. Pihnastyi O. M. Using pde model and system dynamics model for describing multi-operation production lines / O. Pihnastyi, D. Yemelianova, D. Lysytsia // EasternEuropean Journal of Enterprise Technologies. – 2020. – №4 (4(106)). – P. 54–60. DOI: 10.15587/17294061.2020.210750

21. Takeda H. The Synchronized Production System / H. Takeda. – London : Kogan Page Ltd, 2006. –264 p.

22. Chankov S. Influencing factors of synchronization in manufacturing systems / S. Chankov, M. Hütt, J. Bendul // International Journal of Production Research. – 2018. – № 56 (14). – P. 4781–4801. DOI: 10.1080/00207543.2017.1400707

23. Kempf K. Simulating semiconductor manufacturing systems: successes, failures and deep questions  / K. Kempf // Proceedings of the 1996 Winter Simulation Conference, Institute of Electrical and Electronics Engineers. – Piscataway, New Jersey. – 1996. – P. 3–11. DOI: 10.1109/WSC.1996.873254

24. Bitran G. Hierarchical production planning: a two-stage system / G. Bitran, E. Haas, A. Hax // Operations Research. – 1982. – № 30. – P. 232–251. DOI: 10.1287/opre.30.2.232

25. Pihnastyi O. M. Stochastic equation of the technological process / O. M. Pihnastyi, V. D. Khodusov // International Conference on System analysis & Intelligent computing. – 2018. – P.64–67. DOI:10.1109/SAIC.2018.8516833